An important problem in the theory of Lagrangian submanifolds is to find non-trivial examples of Lagrangian submanifolds in complex Euclidean spaces with some given special geometric properties. In this article, we provide a new method to construct Lagrangian surfaces in the complex Euclidean plane C2 by using Legendre curves in S3(1) $\subset$ C2. We also investigate intrinsic and extrinsic geometric properties of the Lagrangian surfaces in C2 obtained by applying our construction method. As an application we provide some new families of Hamiltonian minimal Lagrangian surfaces in C2 via our construction method.